Smarandachely antipodal signed digraphs by Ryan Elias

Scientia Magna Vol. 6 (2010), No. 3, 84-88

Smarandachely antipodal signed digraphs P. Siva Kota Reddy† , B. Prashanth† and M. Ruby Salestina‡ † Department of Mathematics Acharya Institute of Technology, Bangalore 560090, India ‡ Department of Mathematics Yuvaraja’s College, University of Mysore, Mysore 570005, India E-mail: pskreddy@acharya.ac.in bbprashanth@yahoo.com salestina@rediffmail.com Abstract A Smarandachely k-signed digraph (Smarandachely k-marked digraph) is an ordered pair S = (D, σ) (S = (D, µ)) where D = (V, A) is a digraph called underlying digraph of S and σ : A → (e1 , e2 , ..., ek ) (µ : V → (e1 , e2 , ..., ek )) is a function, where each ei ∈ {+, −}. Particularly, a Smarandachely 2-signed digraph or Smarandachely 2-marked digraph is called abbreviated a signed digraph or a marked digraph. In this paper, we define the Smarandachely → − antipodal signed digraph A (D) of a given signed digraph S = (D, σ) and offer a structural characterization of antipodal signed digraphs. Further, we characterize signed digraphs S for → − → − → − which S ∼ A (S) and S ∼ A (S) where ∼ denotes switching equivalence and A (S) and S are denotes the Smarandachely antipodal signed digraph and complementary signed digraph of S respectively. Keywords Smarandachely k-signed digraphs, Smarandachely k-marked digraphs, balance, switching, Smarandachely antipodal signed digraphs, negation.

§1. Introduction For standard terminology and notion in digraph theory, we refer the reader to the classic text-books of Bondy and Murty [1] and Harary et al.[3] ; the non-standard will be given in this paper as and when required. A Smarandachely k-signed digraph (Smarandachely k-marked digraph) is an ordered pair S = (D, σ) (S = (D, µ)) where D = (V, A) is a digraph called underlying digraph of S and σ : A → (e1 , e2 , ..., ek ) (µ : V → (e1 , e2 , ..., ek )) is a function, where each ei ∈ {+, −}. Particularly, a Smarandachely 2-signed digraph or Smarandachely 2-marked digraph is called abbreviated a signed digraph or a marked digraph. A signed digraph is an ordered pair S = (D, σ), where D = (V, A) is a digraph called underlying digraph of S and σ : A → {+, −} is a function. A marking of S is a function µ : V (D) → {+, −}. A signed digraph S together with a marking µ is denoted by Sµ . A signed digraph S = (D, σ) is balanced if every semicycle of S is positive (Harary et al.[3] ). Equivalently, a signed digraph is balanced if every semicycle has an even number of negative arcs. The following characterization of balanced signed digraphs is obtained by E. Sampathkumar et al.[5] .

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Proposition 1.1.[5] A signed digraph S = (D, σ) is balanced if, and only if, there exist a → in S satisfies σ(− → = µ(u)µ(v). marking µ of its vertices such that each arc − uv uv) Let S = (D, σ) be a signed digraph. Consider the marking µ on vertices of S defined as follows: each vertex v ∈ V , µ(v) is the product of the signs on the arcs incident at v. → ∈ D, σ 0 (− → = Complement of S is a signed digraph S = (D, σ 0 ), where for any arc e = − uv uv) µ(u)µ(v). Clearly, S as defined here is a balanced signed digraph due to Proposition 1.1. In

[5]

, the authors define switching and cycle isomorphism of a signed digraph as follows:

Let S = (D, σ) and S 0 = (D0 , σ 0 ), be two signed digraphs. Then S and S 0 are said to be isomorphic, if there exists an isomorphism φ : D → D0 (that is a bijection φ : V (D) → V (D0 ) −−−−−→ → is an arc in D then − → such that if − uv φ(u)φ(v) is an arc in D0 ) such that for any arc − e ∈ D, − → − → 0 σ( e ) = σ (φ( e )). For switching in signed graphs and some results involving switching refer the paper [4] . Given a marking µ of a signed digraph S = (D, σ), switching S with respect to µ is → of S 0 by µ(u)σ(− → the operation changing the sign of every arc − uv uv)µ(v). The signed digraph obtained in this way is denoted by Sµ (S) and is called µ switched signed digraph or just switched signed digraph. Further, a signed digraph S switches to signed digraph S 0 (or that they are switching equivalent to each other), written as S ∼ S 0 , whenever there exists a marking of S such that Sµ (S) ∼ = S0. Two signed digraphs S = (D, σ) and S 0 = (D0 , σ 0 ) are said to be cycle isomorphic, if there exists an isomorphism φ : D → D0 such that the sign σ(Z) of every semicycle Z in S equals to the sign σ(φ(Z)) in S 0 . Proposition 1.2.[4] Two signed digraphs S1 and S2 with the same underlying graph are switching equivalent if, and only if, they are cycle isomorphic.

§2. Smarandachely antipodal signed digraphs In [2] , the authors introduced the notion antipodal digraph of a digraph as follows: For a − → − → digraph D = (V, A), the antipodal digraph A (D) of D = (V, A) is the digraph with V ( A (D)) = − → V (D) and A( A (D)) = {(u, v) : u, v ∈ V (D) and dD (u, v) = diam(D)}. − → We extend the notion of A (D) to the realm of signed digraphs. In a signed digraph S = (D, σ), where D = (V, A) is a digraph called underlying digraph of S and σ : A → {+, −} is − → − → a function. The Smarandachely antipodal signed digraph A (S) = ( A (D), σ 0 ) of a signed digraph − → S = (D, σ) is a signed digraph whose underlying digraph is A (D) called antipodal digraph and Y → → in − sign of any arc e = − uv A (S), σ 0 (e) = µ(u)µ(v), where for any v ∈ V , µ(v) = σ(uv). u∈N (v)

Further, a signed digraph S = (D, σ) is called Smarandachely antipodal signed digraph, if − → S ∼ = A (S 0 ), for some signed digraph S 0 . The following result indicates the limitations of − → the notion A (S) as introduced above, since the entire class of unbalanced signed digraphs is forbidden to be antipodal signed digraphs. Proposition 2.1. For any signed digraph S = (D, σ), its Smarandachely antipodal signed graph A(S) is balanced.

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→ → in − Proof. Since sign of any arc e = − uv A (S) is µ(u)µ(v), where µ is the canonical marking − → of S, by Proposition 1.1, A (S) is balanced. − → For any positive integer k, the k th iterated antipodal signed digraph A (S) of S is defined as follows: − →0 − →− → A (S) = S, Ak (S) = A ( A k−1 (S)). − → Corollary 2.2. For any signed digraph S = (D, σ) and any positive integer k, A k (S) is balanced. In [2] , the authors characterized those digraphs that are isomorphic to their antipodal digraphs. − → Proposition 2.3.[2] For a digraph D = (V, A), D ∼ = A (D) if, and only if, D ∼ = Kp∗ . − → − → Proof. First, suppose that D ∼ = A (D). If (u, v) ∈ A then (u, v) ∈ A( A (D)). Therefore, ∗ dD (u, v) = 1 = diam(D). Since Kp is the only digraph of diameter 1, we have D ∼ = Kp∗ . ∗ For the converse, if D ∼ = Kp , then diam(D) = 1 and for every pair u, v of vertices in D, − → − → the distance dD (u, v) = 1. Hence, A (D) ∼ = Kp∗ and D ∼ = A (D). We now characterize the signed digraphs that are switching equivalent to their Smarandachely antipodal signed graphs. − → Proposition 2.4. For any signed digraph S = (D, σ), S ∼ A (S) if, and only if, D ∼ = Kp∗ and S is balanced signed digraph. − → − → Proof. Suppose S ∼ A (S). This implies, D ∼ = A (D) and hence D is Kp∗ . Now, if S is any − → signed digraph with underlying digraph as Kp∗ , Proposition 2.1 implies that A (S) is balanced − → and hence if S is unbalanced and its A (S) being balanced can not be switching equivalent to S in accordance with Proposition 1.2. Therefore, S must be balanced. − → Conversely, suppose that S is an balanced signed digraph and D is Kp∗ . Then, since A (S) − → is balanced as per Proposition 2.1 and since D ∼ = A (D), the result follows from Proposition 1.2 again. Proposition 2.5. For any two signed digraphs S and S 0 with the same underlying digraph, their Smarandachely antipodal signed digraphs are switching equivalent. − → Proposition 2.6.[2] For a digraph D = (V, A), D ∼ = A (D) if, and only if, i) diam(D) = 2. ii) D is not strongly connected and for every pair u, v of vertices of D, the distance dD (u, v) = 1 or dD (u, v) = ∞. In view of the above, we have the following result for signed digraphs: − → Proposition 2.7. For any signed digraph S = (D, σ), S ∼ A (S) if, and only if, D satisfies conditions of Proposition 2.6. − → − → Proof. Suppose that A (S) ∼ S. Then clearly we have A (D) ∼ = D and hence D satisfies conditions of Proposition 2.6. − → Conversely, suppose that D satisfies conditions of Proposition 2.6. Then D ∼ = A (D) by Proposition 2.6. Now, if S is a signed digraph with underlying digraph satisfies conditions of Proposition 2.6, by definition of complementary signed digraph and Proposition 2.1, S and − → A (S) are balanced and hence, the result follows from Proposition 1.2. The notion of negation η(S) of a given signed digraph S defined in [6] as follows: η(S) has the same underlying digraph as that of S with the sign of each arc opposite to that given to it

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in S. However, this definition does not say anything about what to do with nonadjacent pairs of vertices in S while applying the unary operator η(.) of taking the negation of S. Proposition 2.4 & 2.7 provides easy solutions to two other signed digraph switching equivalence relations, which are given in the following results. − → Corollary 2.8. For any signed digraph S = (D, σ), S ∼ A (η(S)). − → Corollary 2.9. For any signed digraph S = (D, σ), S ∼ A (η(S)). Problem. Characterize signed digraphs for which − → i) η(S) ∼ A (S). − → ii) η(S) ∼ A (S). − → For a signed digraph S = (D, σ), the A (S) is balanced (Proposition 2.1). We now examine, − → the conditions under which negation η(S) of A (S) is balanced. − → − → Proposition 2.10. Let S = (D, σ) be a signed digraph. If A (G) is bipartite then η( A (S)) is balanced. − → − → Proof. Since, by Proposition 2.1, A (S) is balanced, if each semicycle C in A (S) contains − → even number of negative arcs. Also, since A (D) is bipartite, all semicycles have even length; − → − → thus, the number of positive arcs on any semicycle C in A (S) is also even. Hence η( A (S)) is balanced.

§3. Characterization of Smarandachely antipodal signed graphs The following result characterize signed digraphs which are Smarandachely antipodal signed digraphs. Proposition 3.1. A signed digraph S = (D, σ) is a Smarandachely antipodal signed digraph if, and only if, S is balanced signed digraph and its underlying digraph D is an antipodal graph. − → Proof. Suppose that S is balanced and D is a A (D). Then there exists a digraph H such − → that A (H) ∼ = D. Since S is balanced, by Proposition 1.1, there exists a marking µ of D such → in S satisfies σ(− → = µ(u)µ(v). Now consider the signed digraph S 0 = (H, σ 0 ), that each arc − uv uv) where for any arc e in H, σ 0 (e) is the marking of the corresponding vertex in D. Then clearly, − → 0 ∼ A (S ) = S. Hence S is an Smarandachely antipodal signed digraph. Conversely, suppose that S = (D, σ) is a Smarandachely antipodal signed digraph. Then − → there exists a signed digraph S 0 = (H, σ 0 ) such that A (S 0 ) ∼ = S. Hence D is the A(D) of H and by Proposition 2.1, S is balanced.

Acknowledgement The first two authors would like to acknowledge and extend their heartfelt gratitude to Sri. B. Premnath Reddy, Chairman, Acharya Institutes, for his vital encouragement and support.

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References [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, MacMillan, London, 1976. [2] G. Johns and K. Sleno, Antipodal graphs and digraphs, Internat. J. Math. & Math. Sci., 16(1993), No. 3, 579-586. [3] F. Harary, R. Norman and D. Cartwright, Structural models: An introduction to the theory of directed graphs, J. Wiley, New York, 1965. [4] R. Rangarajan and P. Siva Kota Reddy, The edge C4 signed graph of a signed graph, Southeast Asian Bull. Math., 34(2010), No. 6, 1077-1082. [5] E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy, Characterization of Line Sidigraphs, Southeast Asian Bull. Math., to appear. [6] P. Siva Kota Reddy, S. Vijay and H. C. Savithri, A Note on Path Sidigraphs, International J. Math. Combin., 1(2010), 42-46.